The dual of a space with the Radon-Nikodým property

Collier, James B.

doi:10.2140/pjm.1976.64.103

Cited by 22 publications

(17 citation statements)

References 5 publications

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“…Since X has the RadonNikodym property it follows from [4] that r is Frechet differentiate on a dense Gg subset D\ in X* . So, to finish the proof of Proposition 3 we show that if r is Frechetdifferentiable at x, then x admits a farthest point in C. C 17(0,1).…”

Section: Proposition 3 Let X Be a Banach Space With The Radon-nikodymentioning

confidence: 99%

Farthest points in W*-compact sets

Deville

Zizler

1988

Bull. Austral. Math. Soc.

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We show that while farthest points always exist in w* -compact sets in duals to RadonNikodym spaces, this is generally not the case in dual Radon-Nikodym spaces. We also show how to characterise weak compactness in terms of farthest points.The purpose of this note is to find under what conditions the Edelstein-AsplundLau results on the existence of farthest points in weakly compact sets can be extended to w* -compact sets.To fix our notation, let C be a norm closed bounded subset of a real Banach space X and x be an element of X. We defineand call r(x) the farthest distance from x to C. Equivalently, r(x) is the radius of the smallest ball of centre x , containing C. The function r is convex as supremum of such functions, and continuous since \r(x) -r(y)\ ^ ||x -y\\, for all x , y £ X . A point z £ C is called a farthest point of C if there exists x £ X such that ||x -z|| = r(x). The existence of a farthest point of C is equivalent to the fact that the set is non-empty. Since it follows that any farthest point of a convex set C in a locally uniformly rotund space is a strongly exposed point of C, the notion of a farthest point is widely used in the study of the extreme structure of sets. In fact, it was the first discovered method of obtaining exposed points in sets [10]. We refer the reader to [3] and [5] for unexplained notions.Extending the results of [6] and [1], Lau showed [9] that if C is a weakly compact set in a Banach space X, then the set D defined above is dense in X and thus, in particular, C has farthest points. Other extensions of the results of [6] and [1] may be found in [11]. The connection between farthest points and strongly exposed points in

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Section: Proposition 3 Let X Be a Banach Space With The Radon-nikodymentioning

confidence: 99%

Farthest points in W*-compact sets

Deville

Zizler

1988

Bull. Austral. Math. Soc.

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“…We know a locally uniformly convexifiable Banach space (i.e., a space admitting an equivalent locally uniformly convex norm), in particular, a separable space, would fail to have the Radon-Nikodým property. But combining Collier's theorem [5] with the main results of this paper we see that a locally uniformly convexifiable space is very close to the RNP. And yet, we could not substitute ''generic Fréchet differentiability'' for ''generic Fréchet differentiability approximating.''…”

mentioning

confidence: 51%

“…The space E is said to be an Asplund space provided every continuous convex function on E is generically Fréchet differentiable [15] (see also [16]); and the dual E* of E is said to be a w*-Asplund space if every w*-lower semicontinuous and norm continuous convex function on E* is generically Fréchet differentiable [5] (see also [1]). …”

Section: If In Addition F Is Bounded On Each Bounded Subset Of E* mentioning

confidence: 99%

“…We should remark that Collier [5] showed E* is a w*-Asplund space if and only if the predual E has the Radon-Nikodým property (RNP). The main theorem presented in this paper further explains that though a locally uniformly convexifiable Banach space E would not have the RNP, it is very ''near'' to having the RNP.…”

Section: If In Addition F Is Bounded On Each Bounded Subset Of E* mentioning

confidence: 99%

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Approximation of Convex Functions on the Dual of Banach Spaces

Cheng

Ruan

Teng

2002

Journal of Approximation Theory

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“…A large number of renowned mathematicians have contributed to the development of research in this area and around these themes. We can mention, among others, Asplund, Collier, Godefroy, Pearce, Bourgin, Bourgain, Namioka, Phelps, Castin, Fabien, Maynard, Lee, ..., see for instance: [3], [4], [6], [7], [8], [9], [12], [15], [16], [18], [20], [23] And the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Regularized Minimization of Convex Functions in Radon Nikodym Space

Saidou¹,

Zineddine²,

Ferrahi³

2018

GLM

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Abstract. Convex non linear optimization problems may not have a solution in innite dimension spaces. The aim of this paper is to formulate some new results in this topic by using "technical regularization" of the objective function. The rst result shows that a non linear convex proper lower semi-continuous function, on a Banach space which have the Radon-Nikodym property, could be minimized by using a small regularization. while the second one shows that this regularization can be chosen as small as required. In addition, application tracks are presented and illustrated by elementary examples.

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The dual of a space with the Radon-Nikodým property

Cited by 22 publications

References 5 publications

Farthest points in W*-compact sets

Farthest points in W*-compact sets

Approximation of Convex Functions on the Dual of Banach Spaces

Regularized Minimization of Convex Functions in Radon Nikodym Space

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